STAT1008 Lecture Notes - Lecture 6: Confidence Interval, Junk Food, Null Hypothesis
6 INFERENCE FOR MEANS AND PROPORTIONS
6.1 DISTRIBUTION OF A SAMPLE PROPORTION
Outline
- Standard error for a sample proportion
- Necessary sample size for CLT
- CLT for sample proportions
SE for p̂
Owned Homes
- The esus reports that, of all the atios oupied housig uits, .% are oed the
occupants
- If we were to take random sample of 100 homes, what would the standard error of phat be?
- “E = p− p / = .− . / = .
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Distribution of p̂
Sufficiently Large n
- A normal distribution can be used to approximate the distribution of p̂ as long as np ≥ and n(1 – p) ≥
10. Value of N depends on what p is. If p is large then n can become smaller.
CLT for p̂
- A normal distribution is a good approximation as long as p ≥ ad – p ≥
6.2 CONFIDENCE INTERVAL FOR A SINGLE PROPORTION
Outline
- Confidence interval for a single proportion
- Determining sample size
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SE for p̂
- Problem: he doig iferee, e dot know p!
- Solution: substitute p̂, our best guess for p
Confidence Interval for p
statistic ± z*. ~ SE
Sin Taxes
- In March 2011, a random sample of 1000 US adults were asked
- Do ou faor or oppose si taes o soda ad juk food?
- 320 adults responded in favor of sin taxes.
- Give a 95% CI for the proportion of all US adults that favor these sin taxes.
- Counts are greater than 10 in each category
- For a 95% confidence interval, z* = 1.96
- 0.32 ± 1.96 x √. − ./ = 0.32 ± 0.029 = (0.291, 0.349)
- We are 95% confident that between 29.1% and 34.9% of US adults favor sin taxes on soda and
junkfood
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Document Summary
The (cid:1006)(cid:1004)(cid:1005)(cid:1004) (cid:272)e(cid:374)sus reports that, of all the (cid:374)atio(cid:374)(cid:859)s o(cid:272)(cid:272)upied housi(cid:374)g u(cid:374)its, (cid:1010)(cid:1009). (cid:1005)% are o(cid:449)(cid:374)ed (cid:271)(cid:455) the occupants. E = p(cid:894)(cid:1005) p(cid:895) / (cid:374) = (cid:1004). (cid:1010)(cid:1009)(cid:1005)(cid:894)(cid:1005) (cid:1004). (cid:1010)(cid:1009)(cid:1005)(cid:895) /(cid:1005)(cid:1004)(cid:1004) = (cid:1004). (cid:1004)(cid:1008)(cid:1012) A normal distribution can be used to approximate the distribution of p as long as np (cid:1005)(cid:1004) and n(1 p) : value of n depends on what p is. If p is large then n can become smaller. A normal distribution is a good approximation as long as (cid:374)p (cid:1005)(cid:1004) a(cid:374)d (cid:374)(cid:894)(cid:1005) p(cid:895) (cid:1005)(cid:1004) 6. 2 confidence inter val for a single pro portion. Problem: (cid:449)he(cid:374) doi(cid:374)g i(cid:374)fere(cid:374)(cid:272)e, (cid:449)e do(cid:374)(cid:859)t know p! Solution: substitute p , our best guess for p. In march 2011, a random sample of 1000 us adults were asked (cid:862)do (cid:455)ou fa(cid:448)or or oppose (cid:858)si(cid:374) ta(cid:454)es(cid:859) o(cid:374) soda a(cid:374)d ju(cid:374)k food? (cid:863) 320 adults responded in favor of sin taxes.
